In this article, the dependent steps of a negative drift random walk are modelled as a two-sided linear process Xn =-μ+∞∑j=-∞ψn-jεj, where {ε, εn; -∞＜ n ＜ +∞}is a sequence of independent, identically distributed random variables with zero mean, μ＞0 is a constant and the coefficients {ψi;-∞＜ i ＜∞} satisfy 0 ＜∞∑j=-∞|jψj| ＜∞. Under the conditions that the distribution function of |ε| has dominated variation and ε satisfies certain tail balance conditions, the asymptotic behavior of P{supn≥0(-nμ+∞∑j=-∞εjβnj) ＞ x}is discussed. Then the result is applied to ultimate ruin probability.